A Semianalytical Approach for Productivity Evaluation of Complex Wells in Multilayered Reservoirs

Summary We present a new approach to solve the problem of complex wells located in layered reservoirs with crossflow. First, the well path is discretized into a carefully chosen series of source points. Second, pressure solution for each source point is obtained in the Laplace domain anywhere in a rectangular reservoir, where crossflow is permitted between layers and with possibly mixed boundary conditions (no-flow or constant pressure). We used the Fourier transforms in the space variables after removing the time variable by the Laplace transformation. The problem is reduced to a matrix product to solve in the transformed domain. The above method is validated through comparisons with known line source solutions and numerical simulations. It can be used to compute the pressure distribution within a well with complex geometry in a stratified bounded reservoir under the assumption of infinite well conductivity. The well can be located anywhere in the reservoir. Boundaries can be no-flow or constant pressure. Last, variable skins along the well path can be imposed. An application on a field located in the deep offshore of the Gulf of Guinea demonstrates the great value for well design optimization in the context of complex heterogeneous reservoirs where classical analytical approaches are typically inappropriate. Introduction Recent drilling technology advances have led the oil industry to use wells with complex geometry (e.g., multilaterals) for performance improvement, particularly in the context of low-productivity reservoirs or in adverse reservoir conditions such as deep-offshore. At present, evaluating productivity of such complex wells remains a difficult task. Indeed, available analytical approaches do not allow modeling complex well geometry in heterogeneous reservoirs. An extensive literature deals with horizontal wells 1–6 in a single, homogeneous layer. Recently, several authors have considered horizontal wells in multilayered reservoirs. For this type of problem it is almost impossible to obtain a single pressure point that approximates the infinite-conductivity solution for all times. Kuchuk 7 presented a new general method to solve transient-pressure problems in a lateral composite reservoir allowing crossflow between layers. This method is based on the principles of reflection and transmission. To determine an approximate infinite-conductivity solution (constant pressure) at the wellbore, the point-source solution is integrated along the well, and the uniform flux line source so obtained is averaged over the length of the well. The wellbore pressure for a horizontal well in a layered and bounded system is developed by Kuchuk8 using the image method (in the x and y direction) applied to the transient solution. Cinco9,10 investigated steady flow in reservoirs producing through a fully slanted well in an infinite slab reservoir. He obtained the analytical solution by representing the well as an infinite-conductivity source so that the pressure transient is the same along the well. Later, Gommard11 and Lu 12 concentrated on the pressure-transient behavior of slanted wells crossing several layers in multilayered reservoirs with crossflow. Gommard particularly studied an optimum subdivision of the well to calculate the pressure-drop response. Larsen13 proposed a slanted well model in multilayer reservoirs based on the multiple permeability concept. The first limitation of the approach is due to the fact that the perforated zone of the well in each layer is represented by a uniform flux fracture solution. Besides, this model does not allow accounting for the early time radial flow period because of the integration of the governing equation along the z axis. At present, the solutions mentioned above cannot solve the general problem of a complex (nonlinear) well located anywhere in a bounded, multilayered reservoir with crossflow. To attempt solving this general problem, we propose in the following a point-source approach where we keep all the points describing the entire well path in order to obtain a good approximation of transient or pseudo-steady-state (PSS) productivity. A new point-source solution is proposed to solve the pressure-diffusion equation. This method is based on:Fourier transforms together with solutions of transcendental equations, to take into account lateral boundaries,the "quadripole" method to represent the crossflow between layers as well as the top and bottom boundary conditions. This method has been used by several authors14,15 for solving heat-conduction problems, for example, to calculate heat transfer across a two-dimensional plane crack. The paper is organized as follows:recall of the governing equations,description of the point-source solution,description of the complete solution for a complex well,validations against known analytical and numerical solutions,case study on real data, andconclusions. Governing Equations We consider a well with complex geometry in an anisotropic medium bounded, above and below, by horizontal layers with crossflow and laterally by limits with constant pressure or no-flow conditions (respectively, Hk=8 or Hk=0 in Eqs. 2a-2d). The more appropriate coordinate system, due to the boundary surface is a Cartesian one (Fig. 1).